CALIFORNIA INSTITUTE OF TECHNOLOGYControl and Dynamical SystemsCDS 101/110R. M. MurrayFall 2004Homework Set #6 Issued: 8 Nov 04Due: 15 Nov 04All students should complete the following problems:1. Plot the Nyquist and Bode plots for the following systems and compute the gain and phase marginof each. You should annotate your plots to show the gain and phase margin computations. For theNyquist plot, mark the branches corresponding to the following sections of the Nyquist “D” contour:negative imaginary axis, positive imaginary axis, semi-circle at infinity (the curved part of the “D”).(a) Disk drive read head positioning system, using lead compensator (we’ll learn about how to designthese in a week or two):P (s) =1s3+ 10s2+ 3s + 10C(s) = 1000s + 1s + 10(b) Second order system with PD compensator:P (s) =100(100s + 1)(s + 1)C(s) = s + 10Note: you may find it easier to sketch the Nyquist plot from the Bode plot (taking some liberties withthe scale) rather than relying on MATLAB.2. In this problem we will design a PI controller for a cruise control system, building on the exampleshown in class. Using the following transfer function to represent the vehicle and engine dynamics:P (s) =ra/m(s + a)(s + b/m)where r = 5 is the transmission gain (this was labelled k in previous sets), a = 0.2 is the engine lagcoefficient, m = 1000 kg is the mass of the car, and b = 50 N sec/m is the viscous damping coeffient.(a) Consider a proportional controller for the car, u = Kp(r − y). Assuming a unity gain feedbackcontroller, this givesC(s) = Kp.Set Kp= 100 and compute the steady state error, gain and phase margins, and poles/zeros forthe closed loop system. Remember that the gain and phase margins are computed based on theloop transfer function L(s) = P (s)C(s).(b) Consider a proportional + integral controller for the car,C(s) = Kp+Kis.Fill in the following table (make sure to show your work):KpKiStable? Gain Margin Phase Margin Steady State Error Bandwidth500 10050 100050 15 1For each entry in the table, plot the pole zero diagram (pzmap) for the closed loop system andthe step response.Only CDS 110a students need to complete the following additional problems:3. Continuing the previous problem, we will now insert a small amount of time delay into the feedbackpath of the system. A pure time delay of τ seconds satisfies the equationy(t) = u(t − τ )This system is a linear input/output system and it can be shown that its transfer function isG(s) = e−sτ.Unfortunately, MATLAB is not able to perfectly represent a time delay in this form, and so we have touse a Pade approximation, which gives a constant gain transfer function with phase that approximatesa time delay. Using a 2nd order Pade approximation, we can approximate our time delay asG(s) =1 − τs/2 + (τs)2/121 + τs/2 + (τs)2/12This function can be computed using the pade function in MATLAB (although the numerator anddenominator are scaled slightly differently).Assume that there is a time delay of τ seconds, which we will insert between the output of the plantand the controller (as we did in Monday’s lecture).(a) For the case Kp= 50, Ki= 1, insert time delays of τ = 0.25 sec and τ = 0.75 seconds. Usinga Pade approximation, compute the resulting gain and phase margin for each case and computethe overshoot and settling time (2%) for the step responses.(b) Repeat part (a) using Kp= 20, Ki= 0.5 and time delays of 0.75 sec and 1.5 sec.(c) Optional: Plot the Nyquist plot for Kp= 20, Ki= 0.5, τ = 0.75 (with the exact time delay, notthe Pade approximation).4. Consider a simple DC motor with inertial J and damping b. The transfer function isP (s) =1Js2+ bs.For simplicity, choose J = 2, b = 1. In this problem you will design some simple controllers to achievea desired level of performance.(a) Design a proportional control law, C(s) = Kp, that gives stable performance and has a bandwidthof at least 1 rad/s and a phase margin of at least 30 degrees. Plot the step response for the closedloop system using your controller.(b) Consider a proportional + derivative controller (PD) of the formC(s) = Kp+ Kdss + 100b/J.Note that the derivative term (Kd) is slightly modified so that we get a rolloff in controller responseat high frequency. Design a controller (choose Kpand Kd) that gives closed loop bandwidth ω = 10rad/sec and has phase margin of at least 30 degrees. Plot the step response for the closed loopsystem using your controller.2Supplemental problems: If you like, you may do any one problems of the following problems inplace of Problem 4. These problems make use of domain-specific knowledge and so you should only dothem if you are comfortable with that problem area. In addition, these problems are experimental innature and you should ask questions quickly if you get stuck (it might not be your fault!).5. In this problem, we will evaluate the behavior of this circuit, called a Wien bridge oscillator. Thecircuit has both positive feedback (which results in oscillations) and negative feedback (which controlsthe amplitude of these oscillations).-+VVV+-outRCRRRCflamp(a) Calculate (analytically) the transfer function of the positive feedback network,VoutV+(treat theop amp as disconnected). At what frequency isVoutV+= 3 (no phase, resulting in potentiallydestabilizing positive feedback)? Plot the Bode plot of this network, using the values R = 10 kΩand C = 0.01 pF.(b) Calculate the transfer function of the negative feedback network,VoutV−.(c) Now, include the op amp in your analysis, and (analytically) calculateVoutV+. Assume ideal op ampbehavior (i.e., the op amp will make sure that V−= V+). What value ofRlampRfresults in stableoscillations at the frequency calculated in part (a)? There is no gain, so it’s stable, although justbarely. Plot the Bode plot for this transfer function, using the value of Rlampyou just calculated.(d) A real incandescent lamp has a resistance that depends on the RMS (root mean square) current:more current amplitude heats the lamp, resulting in higher resistance. Explain how this ”con-troller” results in oscillations with a stable amplitude at the output of the circuit. (For example,assume we’re in a state with steady oscillations, and then examine what happens if we perturbV+.)Note: This is a common example circuit in electronics textbooks. You may look at such solutions, butplease turn in only what you
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